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Bayesian modeling and analysis of the magnetoencephalography and electroencephalography modalities provide a flexible framework for introducing prior information complementary to the measured data. This prior information is often qualitative in nature, making the translation of the available information into a computational model a challenging task. We propose a generalized gamma family of hyperpriors which allows the impressed currents to be focal and we advocate a fast and efficient iterative algorithm, the iterative alternating sequential algorithm for computing maximum a posteriori (MAP) estimates. Furthermore, we show that for particular choices of the scalar parameters specifying the hyperprior, the algorithm effectively approximates popular regularization strategies such as the minimum current estimate and the minimum support estimate. The connection between priorconditioning and adaptive regularization methods is also pointed out. The posterior densities are explored by means of a Markov chain Monte Carlo strategy suitable for this family of hypermodels. The computed experiments suggest that the known preference of regularization methods for superficial sources over deep sources is a property of the MAP estimators only, and that estimation of the posterior mean in the hierarchical model is better adapted for localizing deep sources.

We consider the problem of restoring an image from a noisy blurred copy of it with the additional qualitative information that the image contains sharp discontinuities whose sizes and locations are unknown. The flexibility of the Bayesian imaging framework is particularly convenient in the presence of such qualitative, rather than quantitative, information. By using a non-stationary Markov model with the variance of the innovation process also unknown, it is possible to take advantage of the qualitative prior information, and Bayesian techniques can be applied to estimate simultaneously the unknown and the prior variance. Here we present a unified approach to Bayesian signal processing and imaging, and show that with rather standard choices of hyperpriors we obtain some classical regularization methods, including the TV and the Perona-Malik regularization, as special cases. The application of Bayesian hyperprior models to imaging applications requires a careful organization of the computations to overcome the challenges coming from the large dimensionality. We explain how the computation of MAP estimates within the proposed Bayesian framework can be made very efficiently by a judicious use of Krylov iterative methods solutions and priorconditioners. The Bayesian approach, unlike deterministic estimation methods which produce a single solution image, provides a very natural way to assess the reliability of single image estimates by a Markov Chain Monte Carlo (MCMC) based analysis of the posterior. Computed examples illustrate the different features and the computational properties of the Bayesian hypermodel approach to imaging.

The restarted GMRES algorithm proposed by Saad and Schultz [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869] is one of the most popular iterative methods for the solution of large linear systems of equations Ax = b with a nonsymmetric and sparse matrix. This algorithm is particularly attractive when a good preconditioner is available. The present paper describes two new methods for determining preconditioners from spectral information gathered by the Arnoldi process during iterations by the restarted GMRES algorithm. These methods seek to determine an invariant subspace of the matrix A associated with eigenvalues close to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved.

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We have developed and implemented a novel mathematical model for simulating transients in surface pH (pHS) and intracellular pH (pHi) caused by the influx of carbon dioxide (CO2) into a Xenopus oocyte. These transients are important tools for studying gas channels. We assume that the oocyte is a sphere surrounded by a thin layer of unstirred fluid, the extracellular unconvected fluid (EUF), which is in turn surrounded by the well-stirred bulk extracellular fluid (BECF) that represents an infinite reservoir for all solutes. Here, we assume that the oocyte plasma membrane is permeable only to CO2. In both the EUF and intracellular space, solute concentrations can change because of diffusion and reactions. The reactions are the slow equilibration of the CO2 hydration-dehydration reactions and competing equilibria among carbonic acid (H2CO3)/bicarbonate ( HCO3-) and a multitude of non-CO2/HCO3- buffers. Mathematically, the model is described by a coupled system of reaction-diffusion equations that—assuming spherical radial symmetry—we solved using the method of lines with appropriate stiff solvers. In agreement with experimental data (Musa-Aziz et al, PNAS 2009, 106:5406–5411), the model predicts that exposing the cell to extracellular 1.5% CO2/10 mM HCO3- (pH 7.50) causes pHi to fall and pHS to rise rapidly to a peak and then decay. Moreover, the model provides insights into the competition between diffusion and reaction processes when we change the width of the EUF, membrane permeability to CO2, native extra-and intracellular carbonic anhydrase-like activities, the non-CO2/HCO3- (intrinsic) intracellular buffering power, or mobility of intrinsic intracellular buffers.

Solving inverse problems with sparsity promoting regularizing penalties can be recast in the Bayesian framework as finding a maximum a posteriori (MAP) estimate with sparsity promoting priors. In the latter context, a computationally convenient choice of prior is the family of conditionally Gaussian hierarchical models for which the prior variances of the components of the unknown are independent and follow a hyperprior from a generalized gamma family. In this paper, we analyze the optimization problem behind the MAP estimation and identify hyperparameter combinations that lead to a globally or locally convex optimization problem. The MAP estimation problem is solved using a computationally efficient alternating iterative algorithm. Its properties in the context of the generalized gamma hypermodel and its connections with some known sparsity promoting penalty methods are analyzed. Computed examples elucidate the convergence and sparsity promoting properties of the algorithm.

SUMMARYMany popular solution methods for large discrete ill-posed problems are based on Tikhonov regularization and compute a partial Lanczos bidiagonalization of the matrix. The computational e ort required by these methods is not reduced signiÿcantly when the matrix of the discrete ill-posed problem, rather than being a general nonsymmetric matrix, is symmetric and possibly indeÿnite. This paper describes new methods, based on partial Lanczos tridiagonalization of the matrix, that exploit symmetry. Computed examples illustrate that one of these methods can require signiÿcantly less computational work than available structure-ignoring schemes.